How to go about cold emailing alumni for non-IB sophomore internships?

I'm currently a freshman at a semi-target, and I've been doing a solid amount of networking (emailing, coffee chats, phone calls, etc...) with BBs for junior year. With these, I've just been asking the standard networking questions and all has been great so far.

My question is, how to I approach networking with smaller companies for a sophomore year internship? I don't mean banks, more like search funds, asset management firms, corp fin roles at companies with alumni, etc..... Basically anywhere that doesn't have a structured intern program as I don't qualify for diversity. I've already compiled a list of people I can reach out to from the alumni database on linkedin, but I don't really know what to say. With IB networking, it's easy to play the long game in that I don't have to actually ask for any internships - it will just reach that point naturally. Should I be more direct with these people I'd reach out to for sophomore internships? Would it help to offer to work unpaid? Or am I supposed to do the "dance" of sorts like thats done with IB?

I'm really lost with this, and I'd appreciate any advice. Thanks in advance.